{"id":"https://openalex.org/W2952253316","doi":"https://doi.org/10.1137/070683386","title":"Beyond Hirsch Conjecture: Walks on Random Polytopes and Smoothed Complexity of the Simplex Method","display_name":"Beyond Hirsch Conjecture: Walks on Random Polytopes and Smoothed Complexity of the Simplex Method","publication_year":2009,"publication_date":"2009-01-01","ids":{"openalex":"https://openalex.org/W2952253316","doi":"https://doi.org/10.1137/070683386","mag":"2952253316"},"language":"en","primary_location":{"id":"doi:10.1137/070683386","is_oa":false,"landing_page_url":"https://doi.org/10.1137/070683386","pdf_url":null,"source":{"id":"https://openalex.org/S153560523","display_name":"SIAM Journal on Computing","issn_l":"0097-5397","issn":["0097-5397","1095-7111"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Computing","raw_type":"journal-article"},"type":"preprint","indexed_in":["arxiv","crossref","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/cs/0604055","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5070281692","display_name":"Roman Vershynin","orcid":null},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Roman Vershynin","raw_affiliation_strings":["vershynin@math.ucdavis.edu#TAB#"],"affiliations":[{"raw_affiliation_string":"vershynin@math.ucdavis.edu#TAB#","institution_ids":[]}]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5070281692"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":0.0,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":{"value":0.35463459,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"39","issue":"2","first_page":"646","last_page":"678"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11830","display_name":"Point processes and geometric inequalities","score":0.9984999895095825,"subfield":{"id":"https://openalex.org/subfields/2604","display_name":"Applied Mathematics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T11830","display_name":"Point processes and geometric inequalities","score":0.9984999895095825,"subfield":{"id":"https://openalex.org/subfields/2604","display_name":"Applied Mathematics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10996","display_name":"Computational Geometry and Mesh Generation","score":0.9973000288009644,"subfield":{"id":"https://openalex.org/subfields/1704","display_name":"Computer Graphics and Computer-Aided Design"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10720","display_name":"Complexity and Algorithms in Graphs","score":0.9927999973297119,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/polytope","display_name":"Polytope","score":0.7403806447982788},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.7017450332641602},{"id":"https://openalex.org/keywords/simplex","display_name":"Simplex","score":0.6547781229019165},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.6526570320129395},{"id":"https://openalex.org/keywords/random-walk","display_name":"Random walk","score":0.643936276435852},{"id":"https://openalex.org/keywords/logarithm","display_name":"Logarithm","score":0.6231372356414795},{"id":"https://openalex.org/keywords/vertex","display_name":"Vertex (graph theory)","score":0.6028552651405334},{"id":"https://openalex.org/keywords/conjecture","display_name":"Conjecture","score":0.6008188724517822},{"id":"https://openalex.org/keywords/linear-programming","display_name":"Linear programming","score":0.5377713441848755},{"id":"https://openalex.org/keywords/time-complexity","display_name":"Time complexity","score":0.49336186051368713},{"id":"https://openalex.org/keywords/simplex-algorithm","display_name":"Simplex algorithm","score":0.4900258779525757},{"id":"https://openalex.org/keywords/inverse","display_name":"Inverse","score":0.45430004596710205},{"id":"https://openalex.org/keywords/binary-logarithm","display_name":"Binary logarithm","score":0.4276072680950165},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.3326954245567322},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.26716163754463196},{"id":"https://openalex.org/keywords/statistics","display_name":"Statistics","score":0.10079324245452881},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.09864908456802368},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.07750493288040161}],"concepts":[{"id":"https://openalex.org/C145691206","wikidata":"https://www.wikidata.org/wiki/Q747980","display_name":"Polytope","level":2,"score":0.7403806447982788},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7017450332641602},{"id":"https://openalex.org/C62438384","wikidata":"https://www.wikidata.org/wiki/Q331350","display_name":"Simplex","level":2,"score":0.6547781229019165},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.6526570320129395},{"id":"https://openalex.org/C121194460","wikidata":"https://www.wikidata.org/wiki/Q856741","display_name":"Random walk","level":2,"score":0.643936276435852},{"id":"https://openalex.org/C39927690","wikidata":"https://www.wikidata.org/wiki/Q11197","display_name":"Logarithm","level":2,"score":0.6231372356414795},{"id":"https://openalex.org/C80899671","wikidata":"https://www.wikidata.org/wiki/Q1304193","display_name":"Vertex (graph theory)","level":3,"score":0.6028552651405334},{"id":"https://openalex.org/C2780990831","wikidata":"https://www.wikidata.org/wiki/Q319141","display_name":"Conjecture","level":2,"score":0.6008188724517822},{"id":"https://openalex.org/C41045048","wikidata":"https://www.wikidata.org/wiki/Q202843","display_name":"Linear programming","level":2,"score":0.5377713441848755},{"id":"https://openalex.org/C311688","wikidata":"https://www.wikidata.org/wiki/Q2393193","display_name":"Time complexity","level":2,"score":0.49336186051368713},{"id":"https://openalex.org/C144521790","wikidata":"https://www.wikidata.org/wiki/Q134164","display_name":"Simplex algorithm","level":3,"score":0.4900258779525757},{"id":"https://openalex.org/C207467116","wikidata":"https://www.wikidata.org/wiki/Q4385666","display_name":"Inverse","level":2,"score":0.45430004596710205},{"id":"https://openalex.org/C63553672","wikidata":"https://www.wikidata.org/wiki/Q581168","display_name":"Binary logarithm","level":2,"score":0.4276072680950165},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.3326954245567322},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.26716163754463196},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.10079324245452881},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.09864908456802368},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.07750493288040161},{"id":"https://openalex.org/C132525143","wikidata":"https://www.wikidata.org/wiki/Q141488","display_name":"Graph","level":2,"score":0.0}],"mesh":[],"locations_count":8,"locations":[{"id":"doi:10.1137/070683386","is_oa":false,"landing_page_url":"https://doi.org/10.1137/070683386","pdf_url":null,"source":{"id":"https://openalex.org/S153560523","display_name":"SIAM Journal on Computing","issn_l":"0097-5397","issn":["0097-5397","1095-7111"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Computing","raw_type":"journal-article"},{"id":"pmh:oai:arXiv.org:cs/0604055","is_oa":true,"landing_page_url":"http://arxiv.org/abs/cs/0604055","pdf_url":"https://arxiv.org/pdf/cs/0604055","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"mag:2952253316","is_oa":true,"landing_page_url":"https://arxiv.org/pdf/cs/0604055v3","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"arXiv (Cornell University)","raw_type":null},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.341.4808","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.341.4808","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://arxiv.org/pdf/cs/0604055v3.pdf","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.341.5370","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.341.5370","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://arxiv.org/pdf/cs/0604055v1.pdf","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.341.8780","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.341.8780","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://arxiv.org/pdf/cs/0604055v2.pdf","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.413.8426","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.413.8426","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www-personal.umich.edu/~romanv/papers/simplex-journal.pdf","raw_type":"text"},{"id":"doi:10.48550/arxiv.cs/0604055","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.cs/0604055","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":null,"raw_source_name":null,"raw_type":"article-journal"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:cs/0604055","is_oa":true,"landing_page_url":"http://arxiv.org/abs/cs/0604055","pdf_url":"https://arxiv.org/pdf/cs/0604055","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2952253316.pdf","grobid_xml":"https://content.openalex.org/works/W2952253316.grobid-xml"},"referenced_works_count":12,"referenced_works":["https://openalex.org/W1524198677","https://openalex.org/W1573820523","https://openalex.org/W1649553227","https://openalex.org/W2014528086","https://openalex.org/W2021495277","https://openalex.org/W2030435702","https://openalex.org/W2034053794","https://openalex.org/W2038953884","https://openalex.org/W2107395148","https://openalex.org/W2117081489","https://openalex.org/W2315557445","https://openalex.org/W2592711915"],"related_works":["https://openalex.org/W3088865259","https://openalex.org/W2158781971","https://openalex.org/W3145564876","https://openalex.org/W2114598978","https://openalex.org/W2937843058","https://openalex.org/W1527978610","https://openalex.org/W2120209114","https://openalex.org/W2971836704","https://openalex.org/W2982575816","https://openalex.org/W3142001427","https://openalex.org/W3199985387","https://openalex.org/W3098872886","https://openalex.org/W2950392293","https://openalex.org/W2010176538","https://openalex.org/W2951216714","https://openalex.org/W2489729120","https://openalex.org/W2134443705","https://openalex.org/W2990982865","https://openalex.org/W2745127509","https://openalex.org/W3035139670"],"abstract_inverted_index":{"The":[0],"smoothed":[1,33,144,209,225,236],"analysis":[2,210],"of":[3,12,19,40,63,68,76,85,97,134,166,170,174,194,204,208,211,234,245,248,269],"algorithms":[4],"is":[5,91,138,165,264],"concerned":[6],"with":[7,56,198],"the":[8,27,45,49,61,66,72,77,83,88,95,103,126,132,143,185,192,195,205,212,221,224,235,243,255,267],"expected":[9,57],"running":[10],"time":[11],"an":[13,41],"algorithm":[14],"under":[15],"slight":[16,37],"random":[17,38,183,249],"perturbations":[18,148],"arbitrary":[20,42,161],"inputs.":[21],"Spielman":[22],"and":[23,71],"Teng":[24],"proved":[25],"that":[26,82,125,259,262],"shadow-vertex":[28],"simplex":[29,46,89,213],"method":[30,47,90,214],"has":[31],"polynomial":[32,59],"complexity.":[34],"On":[35],"a":[36,52,140,156,172,175,180,230,239],"perturbation":[39,78],"linear":[43,162,196,226],"program,":[44],"finds":[48],"solution":[50,193],"after":[51],"walk":[53,86,104,135,222],"on":[54,102,131,136],"polytope(s)":[55],"length":[58,84,133],"in":[60,87,94,266],"number":[62,67,96,268],"constraints":[64,98],"n,":[65],"variables":[69],"d":[70],"inverse":[73],"standard":[74],"deviation":[75],"1/sigma.":[79],"We":[80,113,154,251],"show":[81],"actually":[92],"polylogarithmic":[93,265],"n.":[99],"Spielman-Teng's":[100],"bound":[101],"was":[105],"O(n^{86}":[106],"d^{55}":[107],"sigma^{-30}),":[108],"up":[109],"to":[110,116,184,238],"logarithmic":[111],"factors.":[112],"improve":[114,253],"this":[115],"O(log^7":[117],"n":[118],"(d^9":[119],"+":[120],"d^3":[121],"\\s^{-4})).":[122],"This":[123,188,201,228],"shows":[124],"tight":[127],"Hirsch":[128],"conjecture":[129],"n-d":[130],"polytopes":[137],"not":[139,190],"limitation":[141],"for":[142,159,258],"Linear":[145],"Programming.":[146],"Random":[147],"create":[149],"short":[150],"paths":[151],"between":[152],"vertices.":[153,270],"propose":[155],"randomized":[157],"phase-I":[158],"solving":[160],"programs,":[163],"which":[164],"independent":[167],"interest.":[168],"Instead":[169],"finding":[171],"vertex":[173,181],"feasible":[176,186],"set,":[177],"we":[178],"add":[179],"at":[182],"set.":[187],"does":[189],"affect":[191],"program":[197],"constant":[199],"probability.":[200],"overcomes":[202],"one":[203,216],"major":[206],"difficulties":[207],"--":[215,242],"can":[217],"now":[218],"statistically":[219],"decouple":[220],"from":[223],"program.":[227],"yields":[229],"much":[231],"better":[232],"reduction":[233],"complexity":[237],"geometric":[240],"quantity":[241],"size":[244],"planar":[246],"sections":[247],"polytopes.":[250],"also":[252],"upon":[254],"known":[256],"estimates":[257],"size,":[260],"showing":[261],"it":[263]},"counts_by_year":[],"updated_date":"2026-02-09T09:26:11.010843","created_date":"2025-10-10T00:00:00"}